Riot police were sent in to break up the violence on Wednesday, in which about 350 people were injured.

The opposition is demanding Morsi rescind a decree giving him nearly unrestricted powers and shelve a disputed draft constitution that the assembly passed hurriedly last week.

Al Jazeera's Sherine Tadros, reporting from Cairo, said hundreds of protesters remained in the streets before dawn on Thursday, but that most of Morsi's opponents had retreated.

A small group of opposition activists had been camped outside the palace since Tuesday night, when tens of thousands rallied against the presidential decree.

Supporters of Morsi marched to the palace on Wednesday and tore down the opposition's tents. Witnesses said they threw stones and used clubs to attack demonstrators.

Thirty-two people were arrested on Wednesday, according to a statement from the interior ministry.

Protests spread to other cities, and offices of Morsi's Muslim Brotherhood in Ismailia and Suez were torched.

Both sides blamed the other for starting the clashes: Opposition leaders said Morsi was responsible for the bloodshed, while senior Brotherhood officials accused the opposition of "inciting violence".

'National dialogue'

Morsi did not make any public appearances on Wednesday, but his prime minister, Hisham Qandil, issued a brief statement calling for calm "to give the opportunity for the efforts being made now to begin a national dialogue".

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Hours after the clashes began, a spokesman for the Brotherhood called on protesters to leave. Mahmoud Ghozlan said both sides should "withdraw at the same time and pledge not to return there, given the symbolism of the palace".

Our correspondent said the main message now was a call for dialogue.

"What's really significant and saddening, and very worrying for a lot of people in this country watching their TV screens, is that these are Egyptian civilians fighting Egyptian civilians," she said.

"The country is so divided and polarised. That has been the situation for many months, but it was made all the more intense two weeks ago when Morsi issued this constitutional decree giving himself sweeping powers."

The crisis continues to divide Morsi's government. Three of Morsi's advisers resigned in protest on Wednesday. One of them, Saif Abdelfattah, quit during a live interview with Al Jazeera, blaming the "mummified" political culture in Egypt for his departure.

Two other top Morsi aides resigned last week, including Samir Morcos, a prominent Coptic Christian scholar.

The decree issued by Morsi on November 22 barred the courts from dissolving the controversial 100-member constituent assembly which has been drafting a new constitution. A final draft of the document was sent to Morsi last week, and it is scheduled to face a public referendum on December 15.

'Must be consensus'

While protesters battled outside, Vice President Mahmoud Mekki held a news conference inside the palace and tried to calm the situation.

He urged the opposition to rein in street protests, and said political groups could agree on a plan to amend contentious articles after a new parliament is elected in 2013. He called for "communication between political forces" on the document.

"There must be consensus," he said. "There is real political will to pass the current period and respond to the demands of the public."

A group of prominent opposition leaders, including Mohamed ElBaradei, Hamdeen Sabbahi and Amr Moussa, held a press conference in Cairo on Wednesday night and dismissed Mekki's offer. ElBaradei said the opposition is open to dialogue, but not until Morsi revokes his decree.

All three men blamed Morsi for the violence outside the presidential palace. "He has lost the moral legitimacy to lead Egypt," said Sabbahi, who placed third in the presidential election earlier this year.

But the Muslim Brotherhood quickly turned around and blamed their opponents for the clashes. Gehad el-Haddad, a senior adviser to the Freedom and Justice Party, accused the three opposition leaders of "inciting violence".

"It's very sad to see opposition leaders such as ElBaradei, Hamdeen and Amr Moussa to resort to such levels of talk," he told Al Jazeera. "Such disrespect to the sanctity of peaceful protesting, within the context of democracy, is very alarming."